This reminds me a bit of the Prisoner's Dilemma: http://en.wikipedia.org/wiki/Prisoner%27s_dilemma
Both people can "win" if they trust one another, but will they choose to?
So far as my answer is concerned, well, let's just say I'm not an optimist.

If you haven't read it, the manga Liar Game covers these sorts of scenarios. I read it recently, and found it entertaining.

The only reason to press the red button is if I think the other person would want to kill me. But why would he? He's a complete stranger.

Take an similar situation. The villains put a gun in your hand. They lift your hand with the gun up to a stranger's head. And do the same thing with the stranger.

You're both faced with the same situation. You can either shoot other person, because you want to prevent him from shooting you first. Or you both just put the guns down. Much more logical.

So I choose to put my gun down.

While the second situation is the same in a very basic sense it is also very different in that with the current set up you don't see the other person. You can't get any sort of idea as to what they are thinking. You don't see their death and with it being a simple press of a button you don't even have to believe you killed anyone. That freedom will make people more willing to press the red button and with no sort of clue as to what the other person is thinking the temptation to hit the red button is stronger. It isn't pressing the red button because you want someone to die, it is pressing it so that you guarantee you live. All it takes for you to be on the losing end is someone who thinks their life is more important than whoever is on the other side and for them to not be a risk taker. Then you die pressing a blue button and they go free.